Ferroelectric materials are characterized by hysteresis loops of the volume polarization density P (in C/m2) as a function of the electric field E (in V/m). Current characterization instruments are used to extract simple parameters specific to these loops. Conventionally, the loops are described by a remanent polarization (the polarization in zero field), a maximum polarization, a coercive field and a bias field.
The shape of the hysteresis loops is however very complex and is closely connected with the amplitude of the applied electric field within the material, with the process for producing the material, with the presence of defects within the material, with the measurement frequency, etc. A considerable amount of information is therefore concealed if the determination is limited to only a few parameters.
A theoretical model was proposed by F. Preisach in the article entitled “Über die Magnetische Nachwirkung [On Magnetic Hysteresis]”, Z. Phys. 94, 277–302 (1935), for completely representing the shape of the hysteresis loop via a complete switching density, called the Preisach density.
The precise experimental determination of this Preisach density relies on a mathematical principle disclosed for example in the article entitled “Mathematical models of hysteresis”, IEEE Trans. Magn. MAG-22, 603–608 (1986) by I. D. Mayergoyz.
This determination requires a very large number of loop measurements, and then data processing. At the present time, the measurement methods applied to determining this Preisach density use only a few measurements and rely on an a priori assumption about the form of this density. These are referred to as analytical methods.
A ferroelectric material is generally a good dielectric, the small-signal behavior of which is nonlinear. This behavior is described by the “butterfly” effect of the small-signal capacitance as a function of the quiescent electric field. These effects cannot be modeled by a Preisach density and must therefore be eliminated. The polarization P(E) must therefore be split into two effects (Equation 1), one Prev(E) being locally reversible and the other Pirr(E) being locally irreversible. The locally reversible effects are accessible by measuring the small-signal capacitance. Only the locally irreversible effects can be modeled by a Preisach density. Perfect separation of these two effects cannot be envisaged using current characterization methods:P(E)=Prev(E)+Pirr(E)  (1)
The locally irreversible polarization represents the ferroelectric domain switching state, or in other words the position of the domain walls. Domain wall displacements are subject to a certain dynamic behavior that introduces complex transient phenomena. The transient phenomena are not taken into account in the Preisach model and must therefore be eliminated. Elimination of these transient phenomena is not envisaged in the current characterization methods.